Experimental chattering reduction in sliding mode control using integral resonant signal shaping on a twin rotor aerodynamic system

Abstract

Due to its high resilience to uncertainties and disturbances, Sliding Mode Control (SMC) is frequently employed in nonlinear systems; however, it suffers from an inherent drawback of chattering, resulting in high-frequency control oscillations, increased actuator wear, and energy inefficiency, which often limit its practical application. This paper presents a practical solution to reduce chattering without altering the fundamental structure of SMC, by introducing an Integral Resonant Control (IRC) loop as a signal-shaping stage that preserves robustness while smoothing switching-induced oscillations. The proposed SMC–IRC approach is experimentally validated on a twin rotor aerodynamic system, a strongly coupled nonlinear MIMO platform that emulates helicopter-like dynamics, and its performance is evaluated under both step and sinusoidal tracking tasks in comparison with conventional SMC and Super-Twisting SMC. The results demonstrate that the proposed method maintains comparable tracking accuracy while significantly reducing chattering and control energy; in sinusoidal tracking, it improves actuator command smoothness and reduces control signal variation, while step-response results confirm reduced mechanical stress with only a slight trade-off in transient speed. Overall, the findings indicate that resonance-based signal shaping provides an effective and actuator-friendly enhancement to SMC, offering a simple, computationally efficient solution suitable for robotic and real-time aerospace applications where hardware protection and smooth control action are critical.

Keywords

sliding mode control integral resonant control chattering suppression super-twisting algorithm twin rotor aerodynamic system

1. Introduction

Sliding Mode Control (SMC) is widely used for nonlinear systems due to its robustness against uncertainties and disturbances (Bascetta and Incremona, 2021; Fatemi and Akbarimajd, 2025; Paiva et al., 2019). It ensures accurate tracking by driving system states onto a predefined sliding surface (An et al., 2018; Longfei et al., 2019). However, its main limitation is chattering, caused by high-frequency switching in the control law (Abbasi and Lee, 2018; Lei and Li, 2018). This can lead to actuator wear, excitation of unmodeled dynamics, reduced tracking accuracy, and potential stability issues (Suleiman et al., 2018; Tseng and Chen, 2010; Zhang et al., 2025).

The Twin Rotor Aerodynamic System (TRAS) is a standard benchmark due to its nonlinear, coupled, and sensitive dynamics (Ahmad et al., 2000; I; Gu et al., 2013; NTECO Ltd., 2019). As a result, SMC is widely applied to TRAS, UAVs, and multirotor systems (Yin et al., 2018). Existing approaches include adaptive SMC for nonlinear handling (He et al., 2026; Król, 2024; Kumar et al., 2025; Shen et al., 2025), integral SMC for improved steady-state accuracy (Eltayeb et al., 2022), and terminal SMC for finite-time convergence (Nguyen et al., 2022; Weidong et al., 2015). Despite these advances, effective chattering reduction remains essential for stability and actuator protection (He et al., 2026; Labdai et al., 2020; Zeghlache et al., 2016).

Higher-order sliding mode (HOSM) methods, such as the Super-Twisting (ST-SMC) algorithm, reduce chattering by shifting discontinuities to the derivative of the control signal, resulting in a continuous output (Bkekri et al., 2019; Vijapur et al., 2024; Zhang et al., 2021a; 2021b). While this approach improves smoothness in quad-rotor and TRAS applications (Chen, 2024; Wang and Ji, 2025), its performance can degrade in systems with strong resonant modes or frequency-domain constraints (Zhang et al., 2021a; 2021b). Consequently, achieving a balance between robustness and smoothness still requires intensive gain tuning. Vibration control techniques like Integral Resonant Control (IRC) were originally developed for active damping in flexible cantilever structures and nano-positioning systems (Abdullahi et al., 2014; Díaz et al., 2012; Yoshida et al., 2023). By introducing favorable phase characteristics near resonance, IRC improves damping without compromising low-frequency tracking performance, a synergy successfully demonstrated in precision robotics (Hamza et al., 2026; Omidi and Mahmoodi, 2016; Saeed et al., 2022). However, the use of IRC specifically for chatter mitigation within a Sliding Mode Control (SMC) framework, particularly for coupled MIMO systems like the TRAS, remains largely unexplored. This paper proposes a novel hybrid architecture: SMC–IRC. Building on previous simulation studies of a 3-DOF exoskeleton, this work provides the first experimental validation of the SMC–IRC on a laboratory-scale TRAS platform. The primary objective is to utilize the IRC loop to attenuate high-frequency control components while maintaining robustness and disturbance-rejection properties inherent to SMC.

Three key observations motivate this work. First, while SMC offers fast convergence and robustness, chattering remains a major drawback for nonlinear, cross-coupled systems like the TRAS. Second, higher-order methods like ST-SMC provide smoother signals but require complex tuning and often exhibit residual oscillations. Third, although conventional filters can reduce chattering effects, they frequently compromise tracking performance; in contrast, IRC offers frequency-selective damping, yet its application in nonlinear MIMO sliding mode control systems remains limited. In response, a practical approach, a hybrid SMC–IRC control, is presented in this paper to reduce chattering in SMC while keeping its well-known robustness, and the results are compared with ST-SMC and conventional SMC. The key contributions are:

(1) A practical chattering reduction method that preserves SMC robustness by using an Integral Resonant Control (IRC) loop to smooth switching without modifying the core SMC law.

(2) Incorporation of resonance-based damping into nonlinear control to suppress high-frequency oscillations while maintaining low-frequency tracking performance.

(3) Experimental validation on a TRAS shows that SMC–IRC reduces chattering and control effort while maintaining comparable tracking accuracy against SMC and ST-SMC.

(4) Improved actuator performance through smoother control signals, reducing energy use and mechanical stress for real-time applications.

Overall, SMC–IRC offers a simple, effective way to enhance sliding mode control smoothness without sacrificing robustness.

2. Dynamic modeling of the TRAST

A model that emulates a two-axis UAV is a laboratory-scale INTECO TRAS shown in Figure 1. It has two degrees of freedom: yaw and pitch. Aerodynamic forces caused by the propeller result in strong nonlinear coupling between the two axes and make TRAS a standard benchmark in the evaluation of nonlinear control [12].

Figure 1.

TRAS experimental platform.

2.1. System configuration

The TRAS is a nonlinear MIMO system that is highly coupled between the yaw and the pitch. It is modeled with a reduced-order rigid-body model, ignoring electrical dynamics and saturation. Rather, the robustness of the sliding mode controller is used to deal with uncertainties and nonlinearities. Experiments are used to verify model validity, and the parameters are provided in Table 1. The system has two generalized coordinates: yaw and pitch.

q = [\begin{array}{l} q_{1} \\ q_{2} \end{array}]

(1)

where

q_{1}

and

q_{2}

represent the yaw and pitch angles, respectively.

Table 1.

System parameters.

Parameters	Symbols	Values
Yaw inertia	$J_{y a w}$	0.0242 kg $m^{2}$
Pitch inertia	$J_{p i t c h}$	0.0307 kg $m^{2}$
Yaw damping coefficient	$b_{y a w}$	0.2 nms/rad
Pitch damping coefficient	$b_{p i t c h}$	0.2 nms/rad
Coupling stiffness	$k_{c}$	0.01 nm/rad
Gravity	$g$	9.81 m/ $s^{2}$

The general nonlinear dynamic equation of the TRAS can be written as

M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + D \dot{q} + K q = τ + d (t)

(2)

where

M (q)

is the inertia matrix,

C (q, \dot{q})

represents Coriolis and centrifugal effects,

D

denotes viscous damping,

K

represents cross-coupling stiffness,

τ

is the control torque vector, and

d (t)

is an external disturbance torque.

Cross-coupling stiffness causes motion or disturbances in one axis to induce oscillations in the other, creating a resonant interaction that can be amplified by SMC chattering. $J_{y}$ and $J_{p}$ represent the yaw and pitch inertias, while $B_{y}$ and $B_{p}$ are the viscous damping coefficients. External aerodynamic disturbances are modeled as time-varying sinusoidal inputs applied over a defined time interval.

d (t) = {\begin{cases} [\begin{array}{l} A_{y} \sin (ω_{y} t) \\ A_{p} \sin (ω_{p} t) \end{array}], & 4.0 < t < 6.0 \\ 0, & otherwise \end{cases}

(3)

The disturbance injection near the system’s operational frequencies allows assessment of the controller’s ability to remain on the sliding surface while preventing excitation of resonant coupling modes. The symmetric inertia matrix $M (q)$ is given by

M (q) = [\begin{array}{l} J_{y} & 0 \\ 0 & J_{p} \end{array}]

(4)

The inertia matrix is diagonal, reflecting the principal rotational inertias of the yaw and pitch axes. This assumption suggests that the inertial cross-axis coupling is relatively weak when compared to other dynamic effects, which is consistent with typical TRAS mass distributions that are decoupled.

Similarly, the damping matrix is defined as

D = [\begin{array}{l} B_{y} & 0 \\ 0 & B_{p} \end{array}]

(5)

indicating that viscous damping is primarily localized within each rotational axis.

The cross-coupling stiffness matrix is expressed as

K = [\begin{array}{l} 0 & k_{c} \\ k_{c} & 0 \end{array}]

(6)

where

k_{c}

represents the coupling stiffness coefficient. These model equations dominate the coupling mechanism in the TRAS system, where motion along one axis induces a reflection effect in the other due to aerodynamic interaction and structural linkage.

2.2. State-space representation

For control design and stability analysis, the model is converted into a first-order state-space form. Defining the state vector as

x = {[\begin{array}{l} q_{1} & q_{2} & {\dot{q}}_{1} & {\dot{q}}_{2} \end{array}]}^{T}

(7)

The system can be represented as

\dot{x} = [\begin{array}{l} \dot{q} \\ M^{- 1} (q) (τ - C (q, \dot{q}) \dot{q} - D \dot{q} - K q + d (t)) \end{array}]

(8)

This state-space form is used for the development of the SMC, ST-SMC, and SMC–IRC controllers in the subsequent section.

2.3. Modeling assumptions and limitations

The TRAS model is a reduced-order representation used for control design. The inertia matrix is assumed diagonal, with yaw and pitch inertias, while coupling is mainly captured through the stiffness terms, highlighting inter-axis resonance relevant to chattering analysis.

Coriolis, centrifugal, and higher-order aerodynamic effects are neglected as they have a limited impact under the operating conditions considered. However, the model does not fully capture strong coupling during aggressive maneuvers, actuator nonlinearities, or structural flexibility.

These effects are naturally present in experiments on the real TRAS platform, allowing the controller’s robustness to be evaluated under realistic uncertainties.

3. Control design

This section describes the development of three controllers for the TRAS: conventional SMC, ST-SMC, and the proposed hybrid SMC–IRC.

3.1. Conventional sliding mode control

Let the desired trajectory be $q_{d} = {[q_{1 d}, q_{2 d}]}^{T}$ with derivatives ${\dot{q}}_{d}$ and ${\ddot{q}}_{d}$ . The tracking error and its derivative are defined as

e = q - q_{d}, \dot{e} = \dot{q} - {\dot{q}}_{d}

(9)

The sliding surface is selected as

s = \dot{e} + Λ e

(10)

where

Λ = diag (λ_{1}, λ_{2})

is a positive definite gain matrix.

The SMC control input is composed of nominal/continuous and switching components:

τ = τ_{n m l} + τ_{s w}

(11)

The nominal PID control component gives the primary tracking effort based on the error dynamics, while the discontinuous sliding mode term compensates for uncertainties, disturbances, and unmodeled nonlinearities.

τ_{n m l} = K_{p} e + K_{i} \int e d t + K_{d} \dot{e}

(12)

while the switching term enforces sliding motion,

τ_{s w} = - K_{s} sat (\frac{s}{ϕ})

(13)

where

K_{s}

is a positive gain matrix and

ϕ

defines the boundary layer thickness.

The novelty of this approach is that, in contrast to traditional SMC schemes, where the equivalent control is determined analytically from the plant model, a practical realization is used, where a PID controller is used to approximate the nominal control.

3.2. Super-twisting sliding mode control

The ST-SMC employs the same sliding surface (10) but replaces the discontinuous switching law with a second-order sliding algorithm. The super-twisting control term is given by

τ_{S T} = - k_{1} ∣ s ∣^{\frac{1}{2}} sign (s) + v

(14)

\dot{v} = - k_{2} sign (s)

(15)

where

k_{1}

and

k_{2}

are positive gains. The total ST-SMC control law becomes

τ = τ_{e q} + τ_{S T}

(16)

This formulation produces a continuous control signal and reduces chattering compared with classical SMC.

3.3. Integral Resonant Control

Integral Resonant Control (IRC) is introduced as a frequency-selective damping element. Its transfer function is defined as

G_{I R C} (s) = \frac{k_{i} s}{s^{2} + 2 ζ ω_{r} s + ω_{r}^{2}}

(17)

where

k_{i}

is the constant gain,

ω_{r}

is the resonant frequency, and

ζ

is the damping ratio. Unlike conventional low-pass filtering, IRC provides phase lead near resonance, allowing high-frequency attenuation without significantly degrading low-frequency tracking authority.

3.3.1. Comparison with filtered SMC

Conventional filtered SMC commonly reduces chattering by passing the discontinuous switching signal through a low-pass filter. A typical second-order low-pass filter can be expressed as

G_{L P F} (s) = \frac{ω_{c}^{2}}{s^{2} + 2 ζ_{c} ω_{c} s + ω_{c}^{2}}

(18)

where

ω_{c}

is the cut-off frequency and

ζ_{c}

is the damping ratio. This filter smooths out the high-frequency switching effects, but it is not related to the plant dynamics, and it could cause phase delay within the control bandwidth. Thus, heavy filtering may slow down the tracking response or worsen the transient response.

The IRC filter in (17) is not a standard low-pass filter but a resonance-based damping element targeting the dominant TRAS mode. Both filtered SMC and SMC–IRC reduce high-frequency switching effects, but differently: the low-pass filter limits bandwidth, while IRC shapes the control signal through resonance damping.

3.3.2. Frequency-domain interpretation of IRC

Comparing (17) and (18), the IRC has zero DC gain due to zero at the origin, so it does not affect low-frequency control and preserves tracking performance. In contrast, the second-order low-pass filter attenuates high frequencies but also introduces phase lag, which can degrade transient response.

3.4. Hybrid SMC–IRC controller

In the proposed scheme, the SMC switching signal is processed through the IRC filter before being applied to the plant. The block diagram is shown in Figure 2. The hybrid control law is

τ = τ_{e q} + G_{I R C} (s) τ_{s w}

(19)

Figure 2.

Block diagram of the proposed SMC–IRC controller.

This structure preserves the robustness of SMC while smoothing high-frequency oscillations responsible for chattering.

IRC parameters were tuned to align with the TRAS platform’s primary oscillation bands. The resonant frequency $ω_{r}$ was selected based on the dominant oscillatory behavior that could be observed in the TRAS yaw-pitch coupling dynamics during preliminary open-loop and closed-loop experimental trials. The IRC resonance was tuned close to this region to maximize the damping effect on the switching-induced oscillations while preserving the low-frequency tracking performance. Frequency components corresponding to high-frequency switching oscillations were found in the main coupling band of the system. Thus, the selected value is a compromise between chattering attenuation and transient response preservation.

3.5. Stability analysis

The stability of the proposed SMC–IRC controller is established using Lyapunov theory.

The sliding surface is defined as

s = \dot{e} + Λ e

(20)

Consider the Lyapunov candidate function

V = \frac{1}{2} s^{T} s

(21)

Differentiating with respect to time yields

\dot{V} = s^{T} \dot{s}

(22)

From the closed-loop dynamics with the SMC law,

\dot{s} = - K_{s} sat (\frac{s}{ϕ}) + d (t)

(23)

where

d (t)

represents bounded disturbances satisfying

∥ d (t) ∥ \leq d_{\max}

Substituting (23) into (22) gives

\dot{V} = - s^{T} K_{s} sat (\frac{s}{ϕ}) + s^{T} d (t)

(24)

using the bound

∣ s^{T} d (t) ∣ \leq ∥ s ∥ d_{\max}

\dot{V} \leq - λ_{\min} (K_{s}) ∥ s ∥ + ∥ s ∥ d_{\max}

(25)

thus, if

λ_{\min} (K_{s}) > d_{\max}

(26)

then

\dot{V} < 0

for

s \neq 0

, ensuring the reachability of the sliding surface.

With the IRC filter inserted, the switching signal becomes $G_{I R C} (s) τ_{s w}$ . Since the IRC transfer function (17) is strictly proper and stable for $ζ > 0$ and $ω_{r} > 0$ , it is BIBO stable. Therefore, the filtered signal introduces only a bounded perturbation $ε (t)$ such that

\dot{s} = - K_{s} sat (\frac{s}{ϕ}) + d (t) + ε (t)

(27)

with

∥ ε (t) ∥ \leq ε_{\max}

Following the same Lyapunov argument,

\dot{V} \leq - λ_{\min} (K_{s}) ∥ s ∥ + ∥ s ∥ (d_{\max} + ε_{\max})

(28)

λ_{\min} (K_{s}) > d_{\max} + ε_{\max}

(29)

then

\dot{V} < 0

for

s \neq 0

, ensuring sliding motion is preserved.

Since the reduced-order dynamics on $s = 0$ are stable, the tracking error converges asymptotically:

e (t) \to 0 as t \to \infty

(30)

Thus, the hybrid SMC–IRC controller guarantees closed-loop stability while attenuating high-frequency chattering.

Additionally, given the IRC transfer function $G_{I R C} (s) = \frac{k_{i} s}{s^{2} + 2 ζ ω_{r} s + ω_{r}^{2}}$ , the filtering effect on the switching signal creates a frequency-dependent perturbation. The induced error is defined as $ε (t) = G_{I R C} (s) τ_{s w} (t) - τ_{s w} (t)$ , which in the frequency domain satisfies $ε (j ω) = (G_{I R C} (j ω) - 1) τ_{s w} (j ω)$ . Since the switching control is bounded, $∥ τ_{s w} (t) ∥ \leq K_{s}$ , the perturbation is bounded by

ε_{\max} \leq \sup_{ω} ∣ G_{I R C} (j ω) - 1 ∣ K_{s}

(31)

For the proposed IRC, the magnitude response is $∣ G_{I R C} (j ω) ∣ = \frac{k_{i} ω}{\sqrt{{{(ω_{r}^{2} - ω^{2})}^{2} + (2 ζ ω_{r} ω)}^{2}}}$ , that has attenuation at low frequencies and amplification of resonance at and around $ω_{r}$ . Therefore, the worst-case deviation is at low-frequency or resonant conditions, providing a conservative bound.

ε_{\max} \leq \max (1, ∣ \frac{k_{i}}{2 ζ ω_{r}} - 1 ∣) K_{s}

(32)

This gives a clear parameter-dependent bound to the perturbation term. The magnitude response of the IRC transfer is a bounded value at all frequencies, as the definition of (17) implies. Thus, the perturbation is such that $∣ ε (t) ∣ \leq ε_{\max} = ∥ G (j ω) ∥_{\infty} \cdot ∣ u_{s w} ∣_{\max}$ , where $∥ G (j ω) ∥_{\infty}$ depends on the IRC parameters, namely, the gain $k$ , damping ratio $ζ$ , and resonant frequency $ω_{n}$ .

The IRC introduces frequency-dependent phase effects, but if the resonant frequency $ω_{n}$ is set well above the sliding mode bandwidth; its influence is limited to high-frequency dynamics. The low-frequency sliding behavior remains unaffected, so phase impact within the control bandwidth is negligible. This preserves the reachability condition and ensures closed-loop stability is maintained.

3.5.1. Effect of IRC dynamics on sliding stability

To further clarify the influence of the IRC loop on system stability, the closed-loop system can be interpreted as an augmented dynamic system that includes the internal IRC states. The IRC transfer function defined in (17) can be written in state-space form as

{\dot{x}}_{i r c} = A_{i r c} x_{i r c} + B_{i r c} u_{s w}, u_{f} = C_{i r c} x_{i r c}

(33)

where

u_{s w}

is the switching component of the SMC law and

u_{f}

is the filtered signal applied to the plant.

The augmented closed-loop system, therefore, becomes

\dot{x} = f (x) + g (x) (u_{e q} + u_{f})

{\dot{x}}_{i r c} = A_{i r c} x_{i r c} + B_{i r c} u_{s w}

(34)

Since the IRC transfer function is strictly proper and internally stable for $ζ > 0$ and $ω_{r} > 0$ , the internal states $x_{i r c}$ remain bounded for any bounded switching signal. Because the switching control $u_{s w}$ is bounded by design through the boundary layer parameter $ϕ$ , the filtered output $u_{f}$ is also bounded.

Therefore, the IRC loop acts as a stable dynamic compensator that introduces a bounded perturbation into the sliding dynamics without altering the sliding manifold itself. The Lyapunov inequality becomes

\dot{V} \leq - (η - Δ_{i r c}) ∣ s ∣

(35)

where

Δ_{i r c}

represents the bounded contribution of the IRC dynamics. By selecting the switching gain

η > Δ_{i r c}

, the reachability condition remains satisfied. Hence, the existence and invariance of the sliding surface are preserved, and asymptotic convergence of the tracking error is maintained.

This configuration filters high-frequency chattering without altering the sliding law, maintaining the system’s robust disturbance rejection. The PID and sliding surface gains are detailed in Table 2.

Table 2.

Nominal control gains and sliding surface control gains.

Joint	Control	Nominal control gains (PID)			Sliding surface control gains
Joint	Control	$K_{p}$	$K_{i}$	$K_{d}$	C1	C2	Phi	K1	K2	$ω_{n}$	$ζ$
Yaw	SMC	3.8795	0.2867	14.9898	8.0	7.0	0.05	1.0	—	—	—
	ST-SMC	3.0795	0.1067	9.7898	6.0	5.2	-	1.5	0.35	—	—
	SMC + IRC	3.0795	0.1167	8.7898	8.0	5.2	0.05	1.5	—	1.2	0.5
Pitch	SMC	12.7352	2.800	12.7352	10.5	7.8	0.05	1.0	—	—	—
	ST-SMC	4.5352	1.200	8.0094	6.5	5.5	-	0.8	0.3	—	—
	SMC + IRC	6.5400	1.200	8.0094	10.5	5.5	0.05	1.5	—	1.2	0.5

The controller gains reported in Table 2 were independently tuned for each control strategy to achieve stable tracking performance while minimizing excessive control activity. Some gain values appear numerically similar because the controllers were initialized from the same baseline tuning conditions to ensure a fair comparative evaluation on the same TRAS platform. Minor variations were then introduced according to the specific control structure and chattering suppression mechanism of each controller.

4. Experimental setup and performance metrics

This section describes the experimental platform used to validate the proposed control strategy and defines the performance metrics employed for quantitative comparison.

4.1. Experimental setup

Experimental validation was conducted on a 2-DOF TRAS, which replicates helicopter yaw–pitch dynamics using independent main and tail rotors. The complete experimental configuration is illustrated in Figure 3.

Figure 3.

Experimental setup for implementation of SMC, ST-SMC, and SMC–IRC controllers.

In all experiments, a constant sampling time of 0.002 s (500 Hz) was used to ensure reliable real-time control and data acquisition. Measurement noise, sensor quantization, actuator nonlinearities, and unmodeled disturbances of the TRAS platform, such as encoder resolution limits, motor driver electrical noise, and aerodynamic effects inherently belong to the experimental results. Thus, the findings are realistic operating conditions and display the disturbance-rejection property of the proposed controller.

4.2. Performance metrics

To provide a comprehensive evaluation, three performance indices are used: Root Mean Square Error (RMSE), Chattering Index (CI), and Control Energy Consumption (EN) as in [36].

Tracking accuracy is quantified using the RMSE between the reference trajectory and the measured output:

R M S E = \sqrt{\frac{1}{N} \sum_{k = 1}^{N} {(q (k) - q_{d} (k))}^{2}}

(36)

where

N

is the total number of samples. Lower RMSE values indicate better tracking performance.

Chattering is evaluated using the CI, which measures the total variation of the control signal:

C I = \sum_{k = 1}^{N - 1} ∣ τ (k + 1) - τ (k) ∣

(37)

The control effort is quantified using the energy consumption index:

E N = \sum_{k = 1}^{N} ∣ τ (k) ∣ Δ t

(38)

where

Δ t

is the sampling period. This metric represents the total actuation effort required to achieve the desired tracking performance.

All the experiments were performed with the same initial conditions and controller parameters to be fairly compared. Repeated trials on the TRAS platform showed similar behavior and consistent responses, with only slight differences observed in performance measures. The reported performance indices (RMSE, CI, and EN) were considered over the entire duration of the experiment to represent overall system performance.

5. Results and discussion

This section provides a detailed comparison of three control strategies: SMC, ST-SMC, and the proposed SMC–IRC. The comparison is based on regulating a step input and tracking a sinusoidal trajectory.

5.1. Simulation results

The simulation studies were performed using the TRAS model and used to establish baseline performance trends under idealized conditions before hardware validation.

5.1.1. Step-input tracking performance

Figure 4 presents the yaw and pitch step responses for all three controllers. In each case, the target set-points are reached, confirming that the closed-loop system achieves stabilization in both channels. The transient characteristics, however, reveal notable differences. The conventional SMC and ST-SMC typically deliver faster rise times, but this rapid convergence is accompanied by overshoot and oscillatory settling. Within the TRAS platform, such behavior is expected, as aggressive control action in one axis injects coupling energy into the other, producing small oscillatory exchanges between yaw and pitch. By contrast, the proposed SMC–IRC exhibits a more damped and controlled transient. Importantly, the IRC does not diminish the low-frequency authority required to reach the set-point; rather, it reshapes the switching-induced high-frequency components, yielding a response that is less aggressive in terms of control activity. The proposed SMC–IRC hybrid is different from filtered SMC in that IRC is implemented in the control loop as a dynamic compensator and not used as an external smoothing filter. This allows the reduction of chattering and the dynamic shaping of performance at the same time.

Figure 4.

Step tracking performance comparison for (a) yaw angle (rad) and (b) pitch angle (rad) under SMC, ST-SMC, and SMC–IRC.

5.1.2. Sinusoidal tracking performance

Figure 5 demonstrates the sinusoidal tracking performance that is more difficult to handle than the step input, since it should be controlled continuously without phase lag. The proposed SMC–IRC has good phase and amplitude tracking, but the conventional SMC and ST-SMC have small oscillations. The reason is that sinusoidal inputs never stop driving the system, and high-frequency switching can always inject unwanted energy into resonant and coupling dynamics. The IRC is a frequency-selective damping mechanism that suppresses such components but still achieves the necessary control action. In contrast to step responses, sinusoidal tracking displays behavior in the frequency domain and resonance behavior. Even though ST-SMC lowers the discontinuities, high-frequency energy is still added. IRC dampens these elements, leading to smoother and more precise tracking.

Figure 5.

Sinusoidal tracking performance comparison for (a) yaw angle (rad) and (b) pitch angle (rad) under SMC, ST-SMC, and SMC–IRC.

5.1.3. Sliding surface convergence

Figures 6 and 7 depict the behavior of the sliding surface in response to step and sinusoidal action in the yaw and pitch. The main difference between conventional SMC and ST-SMC is that the former displays pronounced oscillations around the sliding surface because of the discontinuous switching, whereas the latter reduces these oscillations. The proposed SMC–IRC gives the smoothest convergence to s = 0. This leads to a smoother actuator torque, lower coupling oscillations between yaw and pitch, greater dynamic performance, and enhanced hardware safety, which proves the appropriateness of SMC–IRC to a practical application.

Figure 6.

Sliding surface convergence for all controllers under step input (simulation).

Figure 7.

Sliding surface convergence for all controllers under sinusoidal input (simulation).

5.2. Experimental results

The simulation trends in the real conditions, such as sensor noise, actuator saturation, and aerodynamic uncertainties, are confirmed by the experimental results. All real-time experiments made use of a fixed sampling time of 0.002 s (500 Hz), which was chosen to suit the TRAS dynamics and controller calculation requirements and minimize the effects of discretization. The inherent noise and measurement disturbances of the real world are included in the results and give a realistic measure of controller performance.

5.2.1. Step-input tracking

Figure 8 compares experimental step responses for yaw and pitch. Although all controllers reach their set-points, standard SMC and ST-SMC exhibit faster rise times at the cost of high overshoot and oscillations. Conversely, the SMC–IRC provides a damped, stable response without abrupt excitation. Real-world hardware factors, like motor dead-zones and noise, exacerbate switching issues, making the SMC–IRC’s reduced switching activity more critical in practice than in simulation.

Figure 8.

Experimental step tracking performance for yaw angle (rad) and pitch angle (rad).

5.2.2. Sinusoidal tracking

Figure 9 shows the experimental sinusoidal tracking results. The proposed SMC–IRC tracks the reference more closely, while SMC and ST-SMC exhibit phase lag and small oscillations. Quantitatively, SMC–IRC achieves the lowest cumulative error (RMSE_sum = 0.0264), outperforming ST-SMC (0.0297) and SMC (0.0278). The IRC loop reduces high-frequency switching effects, preventing unwanted oscillations in the control signal. This leads to smoother response, reduced vibration, and improved tracking performance in periodic motion tasks.

Figure 9.

Experimental sinusoidal tracking performance for yaw angle (rad) and pitch angle (rad).

5.2.3. Experimental control signals

Figures 10 and 11 display experimental control torques. Conventional SMC exhibits a dense high-frequency oscillatory control signal, which may increase actuator stress and reduce hardware reliability. ST-SMC provides some improvement but still shows residual oscillatory behavior. In contrast, the SMC–IRC produces the smoothest control torque, indicating that the IRC loop effectively reshapes the discontinuous control signal and substantially reduces actuator stress.

Figure 10.

Control signals under step input for SMC, ST-SMC, and SMC–IRC (Nm).

Figure 11.

Control signals under sinusoidal input for SMC, ST-SMC, and SMC–IRC (Nm).

In practice, this smoothness is vital for reliability. Unlike simulation, real-world chattering causes motor overheating, driver stress, mechanical wear from torque ripple, and battery drain. By eliminating these issues, the SMC–IRC ensures superior stability and protects electromechanical components.

5.2.4. Sliding surface behavior

Figure 12 (step response) and 13 (sine response) illustrate the experimental convergence of the sliding surface. For robust sliding operation, the system states are expected to reach s = 0 and remain in close proximity despite external disturbances. The SMC–IRC demonstrates a stable reaching phase with smoother convergence, confirming that the incorporation of IRC does not compromise reachability but effectively attenuates oscillatory motion around the manifold.

Figure 12.

Sliding surface convergence under step input.

From an engineering standpoint, a smoother sliding surface translates into fewer control reversals and reduced high-frequency mechanical vibration. This observation not only substantiates the stability arguments but also highlights the actuator protection benefits, reinforcing the practical value of resonance-based damping in sliding mode control (Figure 13).

Figure 13.

Sliding surface convergence under sinusoidal input (experimental).

5.2.5. Quantitative performance comparison and discussion

Table 3 quantitatively compares the three controllers, highlighting trade-offs between accuracy, chattering, and control effort. For step tracking, ST-SMC achieves the lowest RMSE (0.0281), followed by SMC (0.0412) and SMC–IRC (0.0520), showing that SMC–IRC trades slight transient speed for significantly reduced actuator workload. In sinusoidal tracking, SMC–IRC excels with an RMSE of 0.0264, outperforming both SMC (0.0278) and ST-SMC (0.0297). This confirms the IRC filters high-frequency switching without degrading low-frequency authority.

Table 3.

Performance Metrics (Step vs Sine Wave).

Scenario	Controller	RMSE_sum	CI_sum	EN_sum
Step input	SMC–IRC	0.0520	642.97	1.1558
	ST-SMC	0.0281	1545.65	6.8291
	SMC	0.0412	2551.36	21.6176
Sine wave	SMC–IRC	0.0264	186.78	0.3774
	ST-SMC	0.0297	1705.74	7.4238
	SMC	0.0278	2815.40	22.1669

Experimental results confirm these trends. For sinusoidal tracking, the SMC–IRC reduces yaw-axis chattering by 69.5% (1410 to 430 Nm/s) and pitch-axis chattering by 67.7% (930 to 300 Nm/s). Across all inputs, chattering reduction remains above 50%, directly translating to smoother commands, lower mechanical stress, and reduced motor thermal loading. While the ST-SMC offers high transient accuracy, it is associated with excessive control energy consumption and high switching activity. In contrast, the proposed SMC–IRC approach provides a more balanced control strategy by permitting limited transient aggressiveness in exchange for substantially improved reliability, robustness, and energy efficiency. This trade-off is essential for real-world hardware longevity.

These findings indicate that the proposed controller achieves a balanced trade-off between tracking accuracy and control smoothness. The reduction in CI and EN directly translates to decreased actuator stress and improved operational efficiency, which are critical for real-world aerospace and robotic applications.

6. Conclusion

This paper presents a hybrid SMC–IRC technique that effectively mitigates chattering in nonlinear systems by applying resonance-based damping to the control signal. Unlike traditional methods, this approach preserves the robustness and disturbance rejection of Sliding Mode Control without modifying the switching law itself. Experimental validation on a TRAS demonstrates that the SMC–IRC reduces the Chattering Index by over 65% and decreases control energy by an order of magnitude compared to conventional and Super-Twisting SMC. While the increased damping introduces a slight trade-off in transient RMSE for step responses, the resulting control action is significantly smoother, drastically reducing mechanical and thermal stress on the actuators. The proposed integration of frequency-domain resonance damping with nonlinear robust control achieves accurate tracking while enhancing reliability and energy efficiency, making it well-suited for aerospace, robotic, and mechatronic applications.

6.1. Future work

Future directions will examine the proposed SMC-IRC controller in the presence of noise by introducing measurement noise and uncertainties to both simulations and experiments. To test the impact of sensor noise on chattering and control performance, sensor noise such as band-limited Gaussian noise will be added to the measured states.

Both time and frequency domains of the interaction between IRC and noise will be analyzed with focus on noise attenuation, control energy, and potential amplification around the resonance frequency $ω_{r}$ . Tracking error, control effort, and noise sensitivity will be evaluated. Moreover, adaptive or noise-sensitive tuning of the IRC parameters $(k_{i}, ζ, ω_{r})$ will be explored to optimize the trade-off between chattering suppression and noise suppression from measurements.

Footnotes

Acknowledgments

The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number PSAU/2025/01/32871.

ORCID iD

Mukhtar Fatihu Hamza

Funding

The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2025/01/32871).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Abbasi

Lee

(2018) Chattering reduction by using proportional derivative sliding surface in sliding mode control (PDSMC). In: 2018 International Conference on Information and Communication Technology Robotics (ICT-ROBOT). Busan, Korea (South), 1–6. DOI: 10.1109/ICT-ROBOT.2018.8549911.

Abdullahi

Mohamed

Marwan

(2014) Resonant control of a single-link flexible manipulator. Jurnal Teknologi (Sciences & Engineering) 67(5): 1–5. Available at: https://doi.org/10.11113/jt.v67.2840

Ahmad

Chipperfield

Tokhi

(2000) Dynamic modelling and control of a 2-DOF twin rotor multi-input multi-output system. In: Proceedings of the 26th Annual Conference of the IEEE Industrial Electronics Society (IECON 2000), 1047–1052.

Wang

Liu

, et al. (2018) An intelligent terminal sliding mode control algorithm with chattering reduction based on particle swarm optimization. In: 2018 10th International Conference on Modelling, Identification and Control (ICMIC). Guiyang, China, 1–6. DOI: 10.1109/ICMIC.2018.8529871.

Bascetta

Incremona

(2021) Design of sliding mode controllers for quadrotor vehicles via flatness-based feedback and feedforward linearization strategies. In: 2021 European Control Conference (ECC). Delft, Netherlands, 1622–1627. DOI: 10.23919/ECC54610.2021.9655036.

Bkekri

Benamor

Messaoud

(2019) Adaptive super-twisting sliding mode controller for 2-DOF helicopter. In: 2019 International Conference on Control, Automation and Diagnosis (ICCAD). Grenoble, France, 1–6. DOI: 10.1109/ICCAD46983.2019.9037913.

Chen

(2024) Barrier function-based adaptive super-twisting integral terminal sliding mode control for a quad-rotor UAV. IEEE Access 12: 3992–3999. https://doi.org/10.1109/ACCESS.2023.3349271

Díaz

Pereira

Reynolds

(2012) Integral resonant control scheme for canceling human-induced vibrations in light-weight pedestrian structures. Structural Control and Health Monitoring 19(1): 55–69. https://doi.org/10.1002/stc.423

Eltayeb

Rahmat

Basri

MAM

, et al. (2022) Integral adaptive sliding mode control for quadcopter UAV under variable payload and disturbance. IEEE Access 10: 94754–94764. https://doi.org/10.1109/ACCESS.2022.3203058

10.

Fatemi

Akbarimajd

(2025) Adaptive sliding mode control for quadrotor UAVs under disturbances using multi-layer perceptron. IEEE Access 13: 45518–45526. https://doi.org/10.1109/ACCESS.2025.3549356

11.

Petkov

Konstantinov

(2013) Robust control of a twin-rotor aerodynamic system Robust Control Design with MATLAB®. London: Springer, pp. 491–517. DOI: 10.1007/978-1-4471-4682-7_18.

12.

Hamza

Abdullahi

Alqahtani

, et al. (2026) Hybrid sliding mode control with integral resonant control for chattering reduction in a 3-DOF lower-limb exoskeleton rehabilitation. Applied Sciences 16(1): 410. https://doi.org/10.3390/app16010410

13.

Zhang

Yang

, et al. (2026) Deterministic learning-based fast terminal sliding mode control of a 2-DOF helicopter system. IEEE Transactions on Aerospace and Electronic Systems 62: 967–976. https://doi.org/10.1109/TAES.2025.3626307

14.

INTECO Ltd. (2019) Two-Rotor Aerodynamic System (TRAS) User Manual. Kraków: INTECO Ltd.

15.

Król

(2024) Discrete quasi-sliding mode control of a two-rotor aerodynamic system. In: 2024 28th International Conference on Methods and Models in Automation and Robotics (MMAR). Poland, 47–52. DOI: 10.1109/MMAR62187.2024.10680823.

16.

Kumar

Yadav

, et al. (2025) PDD2 sliding mode control of a 2-DOF helicopter system. In: 2025 7th International Conference on Energy, Power and Environment (ICEPE), Sohra (Cherrapunjee). India, 1–5. DOI: 10.1109/ICEPE65965.2025.11139384.

17.

Labdai

Chrifi-Alaoui

Drid

, et al. (2020) Real-time implementation of an optimized fractional sliding mode controller on the quanser-aero helicopter. In: 2020 International Conference on Control, Automation and Diagnosis (ICCAD). Paris, France, 1–6. DOI: 10.1109/ICCAD49821.2020.9260546.

18.

Lei

(2018) An adaptive quasi-sliding mode control without chattering. In: 2018 37th Chinese Control Conference (CCC). Wuhan, China, 1018–1023. DOI: 10.23919/ChiCC.2018.8483672.

19.

Longfei

Yuping

Jigui

, et al. (2019) Fuzzy sliding mode control of permanent magnet synchronous motor based on the integral sliding mode surface. In: 2019 22nd International Conference on Electrical Machines and Systems (ICEMS). Harbin, China, 1–6. DOI: 10.1109/ICEMS.2019.8921882.

20.

Nguyen

Moon

(2022) Continuous nonsingular terminal sliding-mode control with integral-type sliding surface for disturbed systems: application to attitude control for quadrotor UAVs under external disturbances. IEEE Transactions on Aerospace and Electronic Systems 58(6): 5635–5660. https://doi.org/10.1109/TAES.2022.3177580

21.

Omidi

Mahmoodi

(2016) Nonlinear integral resonant controller for vibration reduction in nonlinear systems. Acta Mechanica Sinica 32: 925–934. https://doi.org/10.1007/s10409-016-0577-z

22.

Paiva

Gomez-Redondo

Rodas

, et al. (2019) Cascade first and second order sliding mode controller of a QuadRotor UAV based on exponential reaching law and modified super-twisting algorithm. In: Workshop on Research, Education and Development of Unmanned Aerial Systems (RED UAS), Cranfield, UK, 100–105. https://doi.org/10.1109/REDUAS47371.2019.8999711

23.

Saeed

Mohamed

Elagan

, et al. (2022) Integral resonant controller to suppress the nonlinear oscillations of a two-degree-of-freedom rotor active magnetic bearing system. Processes 10(2): 271. https://doi.org/10.3390/pr10020271

24.

Shen

Yan

, et al. (2025) Robust fixed-time sliding mode attitude control for a 2-DOF helicopter subject to input saturation and prescribed performance. IEEE Transactions on Transportation Electrification 11(1): 1223–1233. https://doi.org/10.1109/TTE.2024.3402316

25.

Suleiman

Mu’azu

Zarma

, et al. (2018) Methods of chattering reduction in sliding mode control: a case study of ball and plate system. In: 2018 IEEE 7th International Conference on Adaptive Science & Technology (ICAST). Accra, Ghana, 1–8. DOI: 10.1109/ICASTECH.2018.8506783.

26.

Tseng

M.-L

Chen

M.-S

(2010) Chattering reduction of sliding mode control by low-pass filtering the control signal. Asian Journal of Control 12(3): 392–398. https://doi.org/10.1002/asjc.195

27.

Vijapur

Jungade

Thomas

(2024) A comparative analysis between sliding mode control and super twisting sliding mode control applied on a quadcopter. In: 2024 IEEE International Conference on Electronics, Computing and Communication Technologies (CONECCT). Bangalore, India, 1–6. DOI: 10.1109/CONECCT62155.2024.10677315.

28.

Wang

(2025) A passive fault tolerance strategy for super-twisting algorithm-based sliding mode control with barrier function for quadrotor UAVs. IEEE Access 13: 115321–115330. https://doi.org/10.1109/ACCESS.2025.3585408

29.

Weidong

Pengxiang

Changlong

, et al. (2015) Position and attitude tracking control for a quadrotor UAV based on terminal sliding mode control. In: 2015 34th Chinese Control Conference (CCC). Hangzhou, China, 3398–3404. DOI: 10.1109/ChiCC.2015.7260164.

30.

Yin

Cheng

, et al. (2018) Design of fractional-order backstepping sliding mode controller for the quadrotor unmanned aerial vehicles. In: 2018 37th Chinese Control Conference (CCC). Wuhan, China, 697–702. DOI: 10.23919/ChiCC.2018.8484245.

31.

Yoshida

Gobara

Hiruta

, et al. (2023) Integral resonant control preserving the rigid body mode of torsional vibration systems. IFAC-PapersOnLine 56(2): 6344–6350. https://doi.org/10.1016/j.ifacol.2023.10.815

32.

Zeghlache

Bouguerra

Ladjal

(2016) Sliding mode controller using nonlinear sliding surface applied to the 2-DOF helicopter. In: 2016 International Conference on Electrical and Information Technologies (ICEIT). Tangiers, Morocco, 332–337. DOI: 10.1109/EITech.2016.7519614.

33.

Zhang

Chen

Shang

, et al. (2021a) Adaptive super-twisting distributed formation control of multi-quadrotor under external disturbance. IEEE Access 9: 148104–148117. https://doi.org/10.1109/ACCESS.2021.3124334

34.

Zhang

Cheng

, et al. (2021b) Adaptive super twisting sliding mode control for flying-wing UAV. In: 2021 33rd Chinese Control and Decision Conference (CCDC). Kunming, China, 236–241. DOI: 10.1109/CCDC52312.2021.9601405.

35.

Zhang

Xie

(2025) Disturbance observer-based terminal sliding mode control for the training safety improvement in robot-assisted rehabilitation. Intelligent Robotics 5(2): 333–354. https://doi.org/10.20517/ir.2025.17